Symmetry, Integrability and Geometry: Methods and Applications On the N-Solitons Solutions in the Novikov–Veselov Equation

We construct the $N$-solitons solution in the Novikov-Veselov equation from the extended Moutard transformation and the Pfaffian structure. Also, the corresponding wave functions are obtained explicitly. As a result, the property characterizing the $N$-solitons wave function is proved using the Pfaffian expansion. This property corresponding to the discrete scattering data for $N$-solitons solution is obtained in [arXiv:0912.2155] from the $\bar\partial$-dressing method.

was investigated in terms of the Riemann surfaces with some group of involutions and the corresponding Prym Θ-functions [5,10,22,27,28,33,37]. On the other hand, it is known that the Novikov-Veselov hierarchy is a special reduction of the two-component BKP hierarchy [23,36,40] (and references therein). In [23], the authors showed that the Drinfeld-Sokolov hierarchy of D-type is a reduction of the two-component BKP hierarchy using two different types of pseudo-differential operators, which is different from Shiota's point of view [37]. Also, in [26], it is shown that the Tzitzeica equation is a stationary symmetry of the Novikov-Veselov equation. Finally, it is worthwhile to notice that the Novikov-Veselov equation (1) is a special reduction of the Davey-Stewartson equation [20,21].
In [2,6,7,8], the rational solutions and line solitons of the Novikov-Veselov equation (1) are constructed by the ∂-dressing method. To get these kinds of solutions, the scattering datum have to be delta-type and the reality of U also puts some extra constraints on them. In [39], the singular rational solutions are obtained using the extended Moutard transformation (4); however, the non-singular rational solutions are constructed in [4].

N -solitons solutions
In this section, one uses successive iterations of the extended Moutard transformation (4) to construct N -solitons solutions.
To obtain the N -solitons solutions, we assume that V = 0 in (1) and recall that ∂∂ = 1 4 . One considers U = − = 0, i.e., where is non-zero real constant. The general solution of (8) can be expressed as where ν(λ) is an arbitrary distribution and Γ is an arbitrary path of integration such that the r.h.s. of (9) is well defined. Next, using (5) and (9), one can construct the N -solitons solutions. Let's take ν m (λ) = δ(λ − p m ) and ν n (λ) = a n δ(λ − q n ), where p m , a n , q n are complex numbers. Then one defines Then a direct calculation of the extended Moutard transformation (4) can yield The N -solitons solutions are defined by and then where t is fixed. The τ -functions are defined as follows. For simplicity, let's denote (11) and notice that F (−λ) = −F (λ). The τ N is defined as where To get the expansion of (12), we use the following useful formula [14,38] Pf where A and B are m × m matrices and s = [m/2] is the integer part of m/2; moreover, we denote by α c the complementary set of α in the subset {1, 2, 3, . . . , m} which is arranged in increasing order, and |α| = α 1 + α 2 + · · · + α 2r for α = (α 1 , α 2 , . . . , α 2r ). For the case (12), one has where A N and B N are 2N × 2N matrices. Hence by (14) one can have the expansion of (12), i.e., where Here we have utilized the formula that if C is a 2N × 2N matrix with (i, j)-th entry α i −α j α i +α j , then one has the Schur identity [32,36] Next, we illustrate the formula (15) (or 12) with several examples.

The wave functions
In this section, one uses (7) to construct the corresponding wave function of the τ function (15). From (7), one knows that the corresponding wave function of the N -solitons (15) can be written as Using the notations in (11), (12) and (13), we can express ϕ N as But we notice that where (·) means there is no δ mn here when compared with (13). Now, let's compute P (−p 1 , q 1 , −p 2 , q 2 , . . . , −p N , q N , λ) using (14). In this case,

Using (16), a simple calculation yields
where and P j k is defined in (15). We give several examples here.
(1) The one-soliton wave function: Then We remark that This is the one-soliton wave function in [2, p. 9].